Succinct Indices for Path Minimum, with Applications to Path Reporting

Abstract

In the path minimum query problem, we preprocess a tree on n weighted nodes, such that given an arbitrary path, we can locate the node with the smallest weight along this path. We design novel succinct indices for this problem; one of our index structures supports queries in O(α(m,n)) time, and occupies O(m) bits of space in addition to the space required for the input tree, where m is an integer greater than or equal to n and α(m,n) is the inverse-Ackermann function. These indices give us the first succinct data structures for the path minimum problem, and allow us to obtain new data structures for path reporting queries, which report the nodes along a query path whose weights are within a query range. We achieve three different time/space tradeoffs for path reporting by designing (a) an O(n)-word structure with $O(\lg^\epsilon n + occ \cdot \lg^\epsilon n)$ query time, where occ is the number of nodes reported; (b) an $O(n\lg\lg n)$-word structure with $O(\lg\lg n + occ \cdot \lg\lg n)$ query time; and (c) an $O( n \lg^\epsilon n)$-word structure with $O(\lg\lg n + occ)$ query time. These tradeoffs match the state of the art of two-dimensional orthogonal range reporting queries [8] which can be treated as a special case of path reporting queries. When the number of distinct weights is much smaller than n, we further improve both the query time and the space cost of these three results.

This work was supported by NSERC and the Canada Research Chairs Program. Part of the first author’s work was done during his visit to the Department of Computer Science and Engineering, Hong Kong University of Science and Technology.